nLab
Pontrjagin-Thom collapse map

Contents

Idea

Given an embedding of manifolds i:XYi : X \hookrightarrow Y, the Thom collapse map is a useful approximation to its would-be left inverse. It is used to define pushforward of cohomology-classes along ii (“Umkehr maps”). It also appears as the key step in Thom's theorem.

Definition

Component definition in topological spaces

All topological spaces in the following are taken to be compact.

Consider XX and YY two manifolds and

i:XY i \colon X \hookrightarrow Y

an embedding.

Write

Definition

The collapse map (or the Pontrjagin-Thom construction) associated to ii and the choice of tubular neighbourhood ff is

c i:YY/(Yf(N iX))Th(N iX), c_i \colon Y \to Y/(Y - f(N_i X)) \stackrel{\simeq}{\to} Th(N_i X) \,,

where the first morphism is the projection onto the quotient and the second is the canonical homeomorphism to the Thom space of the normal bundle.

Remark

Since in the construction of remark 1 every point of N iXN_i X is associated to a particular point of XX, the collapse map lifts to a map

YX +Th(N iX) Y \longrightarrow X_+ \wedge Th(N_i X)

from YY to the smash product of the Thom space (canonically regarded as a pointed topological space) and the topological space XX with a base point adjoined.

(e.g. Rudyak 98, p. 317)

Example

Of particular interest is the case where YY in the above is a Cartesian space dimX+k\mathbb{R}^{dim X + k} or rather its one-point compactification, the sphere S dimX+kS^{dim X + k}. By the Whitney embedding theorem, for every nn \in \mathbb{N} there exists an kk \in \mathbb{N} such that every manifold XX of dimension nn has an embedding X n+kS n+kX \hookrightarrow \mathbb{R}^{n+k} \to S^{n+k}. In this case the collapse map of def. 1 has the form

S n+kTh(N iX). S^{n+k} \longrightarrow Th(N_i X) \,.

Composing this further with the canonical map N iXEO(k)×O(k) kN_i X \longrightarrow E O(k) \underset{O(k)}{\times} \mathbb{R}^{k} to the universal vector bundle of rank kk yields a map

S n+kMO(k) S^{n+k} \longrightarrow M O(k)

from to the kkth space in the Thom spectrum MOM O. This hence defines an element in the homotopy group π k(MO)\pi_{k}(M O) of the Thom spectrum. Thom's theorem says that all elements in the homotopy groups of MOM O arise this way, and that they retain precisely the information of the cobordism equivalence class of manifolds XX.

In this case the refined Thom collapse map of def. 2 is of the form

S n+kX +Th(N iX). S^{n+k} \longrightarrow X_+ \wedge Th(N_i X) \,.
Remark

The refined map in example 1 lifts to a morphism of spectra

𝕊Σ + XΣ nkTh(N iX) \mathbb{S} \longrightarrow \Sigma_+^\infty X \wedge \Sigma^{-n-k} Th(N_i X)

where 𝕊\mathbb{S} denotes the sphere spectrum and Th(N iX)Th(N_i X) now the Thom spectrum of the normal bundle.

This morphism is the unit of an adjunction which exhibits the suspension spectrum Σ + X\Sigma_+^\infty X as a dualizable object in the stable homotopy category, with dual object Σ nkTh(N iX)\Sigma^{-n-k} Th(N_i X). See at Atiyah duality and at n-duality.

Equivalently, one may proceed as follows. For a framed manifold i.e. a manifold M nM^n with a chosen trivialization of the normal bundle N i(M n)N_i (M^n) in some R n+r\mathbf{R}^{n+r} one has TN i(M n)Σ r(M + n)T N_i(M^n)\cong \Sigma^r(M^n_+) where M + nM^n_+ is the union of M nM^n with a disjoint base point. Identify a sphere S n+rS^{n+r} with a one-point compactification R n+r{}\mathbf{R}^{n+r}\cup \{\infty\}. Then the Pontrjagin-Thom construction is the map S n+rTh(N iX)S^{n+r}\to Th(N_i X) obtained by collapsing the complement of the interior of the unit disc bundle D(N iM n)D(N_i M^n) to the point corresponding to S(N iM n)S(N_i M^n) and by mapping each point of D(N iM n)D(N_i M^n) to itself. Thus to a framed manifold M nM^n one associates the composition

S n+rTh(N iX)Σ rM + nS r S^{n+r}\to Th(N_i X)\cong \Sigma^r M^n_+\to S^r

and its homotopy class defines an element in π n+r(S r)\pi_{n+r}(S^r).

Abstract definition in terms of duality

The following is a more abstract description of Pontryagin-Thom collapse in the stable homotopy theory of sphere spectrum-(∞,1)-module bundles.

Definition

Write

D() Σ + L wheTop𝕊Mod D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod

for the Spanier-Whitehead duality map which sends a topological space first to its suspension spectrum and then that to its dual object in the (∞,1)-category of spectra.

(ABG 11, def 10.3).

Proposition

For XX a compact manifold, let X nX \to \mathbb{R}^n be an embedding and write S nX ν nS^n \to X^{\nu_n} for the classical Pontryagin-Thom collapse map for this situation, and write

𝕊X TX \mathbb{S} \to X^{-T X}

for the corresponding looping map from the sphere spectrum to the Thom spectrum of the negative tangent bundle of XX. Then Atiyah duality produces an equivalence

X TXDX X^{- T X} \simeq D X

which identifies the Thom spectrum with the dual object of Σ + X\Sigma^\infty_+ X in 𝕊Mod\mathbb{S} Mod and this constitutes a commuting diagram

X TX 𝕊 D(X*) DX \array{ && X^{- T X} \\ & \nearrow & \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} &\underset{D(X \to \ast)}{\to}& D X }

identifying the classical Pontryagin-Thom collapse map with the abstract dual morphism construction of prop. 3.

More generally, for WXW \hookrightarrow X an embedding of manifolds, then Atiyah duality identifies the Pontryagin-Thom collapse maps

𝕊X TXW TW \mathbb{S} \to X^{-T X} \to W^{- T W}

with the abstract dual morphisms

𝕊DXDW. \mathbb{S} \to D X \to D W \,.

(ABG 11, prop. 10.5).

Remark

Given now ECRing E \in CRing_\infty an E-∞ ring, then the dual morphism 𝕊DX\mathbb{S} \to D X induces under smash product a similar Pontryagin-Thom collapse map, but now not in sphere spectrum-(∞,1)-modules but in EE-(∞,1)-modules.

EDX 𝕊E. E \to D X \otimes_{\mathbb{S}} E \,.

The image of this under the EE-cohomology functor produces

[DX 𝕊E,E]E. [D X \otimes_{\mathbb{S}} E, E] \to E \,.

If now one has a Thom isomorphism (EE-orientation) [DX 𝕊E,E][X,E] [D X \otimes_{\mathbb{S}} E, E] \simeq [X,E] that identifies the cohomology of the dual object with the original cohomology, then together with produces the Umkehr map

[X,E][DX 𝕊E,E]E [X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E

that pushes the EE-cohomology of XX to the EE-cohomology of the point. Analogously if instead of the terminal map X*X \to \ast we start with a more general map XYX \to Y.

More generally a Thom isomorphism may not exists, but [DX 𝕊E,E][D X \otimes_{\mathbb{S}} E, E] may still be equivalent to a twisted cohomology-variant [X,E] χ[X,E]_{\chi} of [X,E][X,E], namely to [Γ X(χ),E][\Gamma_X(\chi),E], where χ:Π(X)ELineEMod\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod is an (flat) EE-(∞,1)-module bundle on XX and and Γlim\Gamma \simeq \underset{\to}{\lim} is the (∞,1)-colimit (the generalized Thom spectrum construction). In this case the above yields a twisted Umkehr map.

(ABG 10, 9.1)

Properties

Proposition

For given ii all collapse maps for different choices of tubular neighbourhood ff are homotopic.

Proof

By the fact that the space of tubular neighbourhoods (see there for details) is contractible.

The following terms all refer to essentially the same concept:

References

An illustration is given on slide 15

The general abstract formulation in stable homotopy theory is in sketched in section 9 of

and is in section 10 of

with an emphases on parameterized spectra.

Revised on June 7, 2016 02:42:37 by Urs Schreiber (131.220.184.222)